Abstract

Slanted lamellar gratings made of dielectric materials are considered, used in conical diffraction mounts. We extend the modal method for slanted lamellar gratings from classical to conical incidence, develop fully generalized Fresnel matrices, and derive energy conservation relations for these matrices. Using the method, we verified a uniaxial crystal model for slanted lamellar gratings in a homogenization regime, examined the effects of grating symmetry on the maximum reflectance of Fano resonances, and showed that slanted lamellar gratings support Fano resonances despite the homogenization of their other optical properties.

© 2009 Optical Society of America

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References

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  1. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
    [CrossRef]
  2. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
    [CrossRef]
  3. L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
    [CrossRef]
  4. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
    [CrossRef]
  5. J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
    [CrossRef]
  6. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581-2591 (1993).
    [CrossRef]
  7. M. P. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817-1834 (2003).
  8. S. Campbell, L. C. Botten, Ross C. McPhedran, and C. Martijn de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415-2426 (2008).
    [CrossRef]
  9. S. Campbell, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
    [CrossRef]
  10. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553-573 (1993).
    [CrossRef]
  11. R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289-312 (1982).
    [CrossRef]
  12. J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821-837 (1994).
    [CrossRef]
  13. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfelds waves),” J. Opt. Soc. Am. 31, 213-222 (1941).
    [CrossRef]
  14. E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607-619 (1986).
    [CrossRef]
  15. A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083-1104 (1988).
    [CrossRef]
  16. I. Avrutsky, “Guided modes in a uniaxial multilayer,” J. Opt. Soc. Am. A 20, 548-556 (2003).
    [CrossRef]

2008 (1)

2007 (1)

S. Campbell, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

2003 (2)

M. P. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817-1834 (2003).

I. Avrutsky, “Guided modes in a uniaxial multilayer,” J. Opt. Soc. Am. A 20, 548-556 (2003).
[CrossRef]

1994 (2)

J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821-837 (1994).
[CrossRef]

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

1993 (2)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581-2591 (1993).
[CrossRef]

1988 (1)

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083-1104 (1988).
[CrossRef]

1986 (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607-619 (1986).
[CrossRef]

1982 (2)

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289-312 (1982).
[CrossRef]

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

1981 (3)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

1941 (1)

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Avrutsky, I.

Botten, L. C.

S. Campbell, L. C. Botten, Ross C. McPhedran, and C. Martijn de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415-2426 (2008).
[CrossRef]

S. Campbell, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289-312 (1982).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Campbell, S.

S. Campbell, L. C. Botten, Ross C. McPhedran, and C. Martijn de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415-2426 (2008).
[CrossRef]

S. Campbell, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

Craig, M. S.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289-312 (1982).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Davidson, M. P.

M. P. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817-1834 (2003).

Fano, U.

Gaylord, T. K.

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083-1104 (1988).
[CrossRef]

Knoesen, A.

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083-1104 (1988).
[CrossRef]

Lekner, J.

J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821-837 (1994).
[CrossRef]

Li, L.

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581-2591 (1993).
[CrossRef]

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

Martijn de Sterke, C.

S. Campbell, L. C. Botten, Ross C. McPhedran, and C. Martijn de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415-2426 (2008).
[CrossRef]

S. Campbell, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

Mashev, L.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607-619 (1986).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607-619 (1986).
[CrossRef]

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289-312 (1982).
[CrossRef]

McPhedran, R. C.

S. Campbell, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289-312 (1982).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

McPhedran, Ross C.

Miller, J. M.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Moharam, M. G.

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083-1104 (1988).
[CrossRef]

Nevière, M.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289-312 (1982).
[CrossRef]

Noponen, E.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607-619 (1986).
[CrossRef]

Sanda, P. N.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Sheng, P.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Taghizadeh, M. R.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Turunen, J.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Vasara, A.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

J. Lightwave Technol. (1)

A. Knoesen, T. K. Gaylord, and M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083-1104 (1988).
[CrossRef]

J. Mod. Opt. (2)

M. P. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817-1834 (2003).

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Acta (5)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607-619 (1986).
[CrossRef]

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289-312 (1982).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

Opt. Commun. (1)

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Phys. Rev. B (1)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Pure Appl. Opt. (1)

J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821-837 (1994).
[CrossRef]

Waves Random Complex Media (1)

S. Campbell, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Slanted diffraction grating and the coordinate system used throughout this paper. The vector v is perpendicular to the plane of incidence (drawn with dashed lines); the polarization angle δ is measured from the vector v in a plane perpendicular to the incident field’s wave vector k to the electric field vector E . The incident field’s incident direction is defined by the angle γ, measured from the positive x-axis in the x z -plane, and the angle ϕ, measured from the positive y-axis to the wave vector k . The angle θ measures the slant of the grating.

Fig. 2
Fig. 2

Structural parameters for the grating considered in Fig. 1. The grating thickness is h, d is the period, f is the fill fraction, ϵ { a , b , 1 , 2 } are permittivities, and θ is the slant angle. Unless stated otherwise, ϵ a = ϵ b = 1 . The incident wave vector is k with incident polar angle ϕ, and conical angle γ (not shown).

Fig. 3
Fig. 3

One vertical period of the slanted lamellar grating is shown. Vectors c 1 ± denote modal amplitudes of basis 1 in region 1, and vectors c 2 ± denote modal amplitudes of basis 2 in region 2. R 12 and T 12 denote Fresnel reflection and transmission matrices.

Fig. 4
Fig. 4

Basis expansion coefficients in air: g ± , an infinitesimal lamellar layer: c ± , and in a semi-infinite slanted lamellar region: d ± .

Fig. 5
Fig. 5

(a) Two capacitor plates induce an electric field across the structure. (b) The periodic refractive index profile along the y axis resulting from large slant angles is illustrated; (c) The optic axis n = ( cos θ , sin θ , 0 ) is shown for a grating of intermediate slant angle in the quasistatic limit λ d , and (d) shows the coordinate system.

Fig. 6
Fig. 6

Reflectance versus the conical angle γ ( ° ) , for a fixed polar angle ϕ = 30 ° . The solid line is the reflectance of a uniaxial crystal under TE incidence, the dashed line is the reflectance for TM incidence, and the circles and squares are the numerical reflectance of a lamellar grating under TE and TM incidence, respectively. The parameters are: ϵ 1 = 9 , ϵ 2 = 1 , λ = 1.1 d , h = 0.4 d , θ = 89 ° , and f = 0.5 .

Fig. 7
Fig. 7

Reflectance as a function of the polar angle ϕ ( ° ) , with conical angle γ = 0 ° (to lowest order the results here are unaffected by the conical angle, as shown in Fig. 6). The solid curve is the reflectance of a uniaxial crystal under TM incidence, the dashed curve for TE incidence, and the circles and squares are the numerical reflectance for TE and TM incidence, respectively, for a grating with parameters: ϵ 1 = 9 , ϵ 2 = 1 , λ = 1.1 d , h = 0.4 d , θ = 89 ° , and f = 0.5 .

Fig. 8
Fig. 8

Reflectance as a function of the slant angle θ ( ° ) . Solid curve: uniaxial crystal reflectance for conical incidence ( ϕ = 25 ° , γ = 25 ° ); squares: reflectance of a slanted lamellar grating. Grating parameters: ϵ 1 = 9 , ϵ 2 = 1 , H = 5 d , f = 0.5 , TM incidence, and λ = 10.8 d .

Fig. 9
Fig. 9

Symmetric chevron grating. The asymmetric grating we consider has h 1 = ( 1 + s ) h , h 2 = ( 1 s ) h where s is the asymmetry parameter. Continuity of the refractive index at the horizontal dashed line is enforced.

Fig. 10
Fig. 10

Maximum reflectivity as a function of the asymmetry parameter s. (b) Fano resonance spectra for s = 0.99 (solid curve), and s = 0 (dashed curve). The grating parameters are: ϵ 1 = 3.4 2 , ϵ 2 = 1 , f = 0.95 , d = 1 , h = 0.3775 , θ = 40 ° , and the incident angles for the TE polarized incident field are ϕ = 10 ° , γ = 0 ° .

Fig. 11
Fig. 11

Reflectance as a function of the slant angle θ ( ° ) for a fixed incident in plane angle of ϕ = 17 ° and conical angle γ = 5 ° . The dashed curve is for a slanted lamellar layer, and the solid curve is the reflectance of a symmetric chevron grating. Grating parameters: ϵ 1 = 5 , ϵ 2 = 1 , h = d , f = 0.5 , λ d = 1.1 , δ = 0 ° .

Fig. 12
Fig. 12

Dashed curve is the reflectance of a slanted lamellar layer as a function of the slant angle θ ( ° ) , and the solid curve is the reflectance of a symmetric chevron grating. The in-plane angle is fixed at ϕ = 17 ° and the conical angle is γ = 5 ° . Grating parameters as in Fig. 11.

Equations (96)

Equations on this page are rendered with MathJax. Learn more.

E = m l m 1 2 ( c m + e i μ m y + c m e i μ m y ) N m E ,
y ̂ × K = m l m 1 2 ( c m + e i μ m y c m e i μ m y ) y ̂ × N m K ,
0 d ( N m E ) * y ̂ × N n K d x = l m δ m n ,
l m = { 1 for propagating modes ± i for evanescent modes . }
L 1 2 ( c 1 + + c 1 ) = J L 1 2 ( c 2 + + c 2 ) ,
J l m = d ( y × N 1 , l K ) * N 2 , m E d x .
J H L 1 2 ( c 1 + + c 1 ) = L 1 2 L * ( c 2 + + c 2 ) ,
G = L 1 2 J L 1 2 ,
H = ( L * ) 1 L 1 2 J H L 1 2 ,
R 12 = ( G H + I ) 1 ( G H I ) ,
T 12 = 2 H ( I + G H ) 1 ,
R 21 = ( I + H G ) 1 ( I H G ) ,
T 21 = 2 G ( I + H G ) 1 .
N 1 , l K ( x ) = N 2 , l K ( x δ x ) ,
= N 2 , l K ( x ) d d x N 2 , l K ( x ) δ x + O ( δ x 2 ) .
J = L V δ x ,
V l m = d ( y × d d x N 2 , l K ( x ) ) * N 2 , m E ( x ) d x .
T = lim N ( I + M δ y ) N = lim N ( I + h N M ) N = exp ( h M ) ,
M = i [ μ 0 0 μ ] + tan θ [ M a M s M s M a ] ,
M a = 1 2 ( M H M G ) ,
M s = 1 2 ( M G + M H ) ,
M H = L 1 2 V L 1 2 ,
M G = ( L * ) 1 L 1 2 V H L 1 2 ,
T = X exp ( h Λ ) X 1 .
X = [ F F F + F + ] ,
R p M ( r ) = 1 d Q p Q p e i Q p r ,
R p E ( r ) = 1 d y ̂ × Q p Q p e i Q p r ,
Q p = ( α p x ̂ + k i , z z ̂ ) ,
α p = k i , x sin ϕ i + 2 π p d ,
E = p = ξ p 1 2 ( e p e i χ p y + e p + e i χ p y ) R p E + ξ p 1 2 ( f p e i χ p y + f p + e i χ p y ) R p M ,
y ̂ × K = p = ξ p 1 2 ( e p e i χ p y e p + e i χ p y ) R p E + ξ p 1 2 ( f p e i χ p y f p + e i χ p y ) R p M ,
χ p = { k 2 Q p 2 Q p 2 < k 2 i Q p 2 k 2 Q p 2 > k 2 } ,
ξ p = χ p k .
g + g + = A ( c + c + ) , A = ζ 1 2 K L 1 2
g ± = [ e ± f ± ] , ζ = [ ξ 0 0 ξ 1 ] , K = [ K E K M ] ,
K p n s = d R p s * N n E d x ,
c + c = B ( g g + ) , B = L 1 2 K H ζ 1 2 .
( c c + ) = ( F F F + F + ) ( d d + ) ,
( c + c + c c + ) = ( E E H H ) ( d d + ) ,
E = F + F + , E = F + F + ,
H = F F + , H = F F + .
c + c = B A ( c + c + ) .
c + c + = b + b + ,
c c + = Y b + Y b + ,
( Y B A ) b = ( B A Y ) b + ,
d = R 21 d + ,
R 21 = E 1 ( Y B A ) 1 ( B A Y ) E .
R 21 = Y 2 Y 1 Y 2 + Y 1 .
R 21 = Y 1 Y 2 Y 2 Y 1 ,
R 23 = E 1 ( Y B A ) 1 ( B A + Y ) E ,
T 21 = A ( Y B A ) 1 ( Y Y ) E ,
T 23 = A ( Y B A ) 1 ( Y Y ) E ,
R 12 = ( A Y 1 B I ) 1 ( A Y 1 B + I ) = A ( Y B A ) 1 ( Y + B A ) A 1 ,
T 12 = 2 E 1 ( Y + B A ) 1 B ,
R 32 = ( A Y 1 B + I ) 1 ( A Y 1 B I ) = A ( Y + B A ) 1 ( Y + B A ) A 1 ,
T 32 = 2 E 1 ( Y + B A ) 1 B .
T 12 T 23 = I R 21 R 23 ,
T 23 T 12 = I R 32 R 12 ,
R sl = R 12 + T 21 P R 23 P ( I R 21 P R 23 P ) 1 T 12 ,
T sl = T 23 P ( I R 21 P R 23 P ) 1 T 12 ,
P = exp ( h Λ f ) ,
P = exp ( h Λ b ) ,
1 2 d y . ( E × K * + E * × K ) = 1 2 m l m ( c m + + c m ) ( c m + ¯ c m ¯ ) + l m ¯ ( c m + ¯ + c m ¯ ) ( c m + c m ) ,
= [ c + c ] H [ I 2 i I 2 ¯ i I 2 ¯ I 2 ] [ c + c ] ,
1 2 d y . ( E × K * + E * × K ) = [ g g + ] H [ I 1 i I 1 ¯ i I 1 ¯ I 1 ] [ g g + ] ,
g ± = [ e ± f ± ] ,
I 1 = [ Re ( η ) 0 0 Re ( η ) ] ,
I 1 ¯ = [ Im ( η ) 0 0 Im ( η ) ] ,
η = diag [ ( ξ p 1 2 ) ( ξ p 1 2 ) * ] .
[ c + c ] = [ I 0 R 21 T 12 ] [ c + g ] ,
[ g g + ] = [ I 0 R 12 T 21 ] [ g c + ] .
[ I R 12 H 0 T 21 H ] [ I 1 i I 1 ¯ i I 1 ¯ I 1 ] [ I 0 R 12 T 21 ] = [ I R 21 H 0 T 12 H ] [ I 2 i I 2 ¯ i I 2 ¯ I 2 ] [ I 0 R 21 T 12 ] .
ϵ o = ϵ 1 c d + ϵ 2 d c d ,
ϵ e = ( c ϵ 1 d + d c ϵ 2 d ) 1 .
β j = k 0 sin ϕ cos γ ± m 2 π d ,
E m , z TE = v m TE , E ( x ) exp ( i μ m TE y ) ,
v m TE , E ( x ) = ( μ m TE ) 1 2 u m TE ( x ) ,
u m ( x ) + ( k 0 2 ϵ ( x ) μ m 2 ) u m ( x ) = 0 ,
E x x ̂ + E y y ̂ = i k 0 2 ϵ ( x ) k z 2 ( k z t E z k 0 z ̂ × K z ) ,
K x x ̂ + K y y ̂ = i k 0 2 ϵ ( x ) k z 2 ( k z t K z + k 0 ϵ ( x ) z ̂ × E z ) ,
K m , z TE = v m TE , K ( x ) exp ( i μ m TE y ) ,
v m TE , K ( x ) = i k z k 0 ( μ m TE ) 1 2 x u m TE ( x ) .
v m TM , K ( x ) = ( μ m TM ) 1 2 u m TM ( x ) ,
v m TM , E ( x ) = i k z k 0 ( μ m TM ) 1 2 1 ϵ ( x ) x u m TM ( x ) ,
ϵ ( x ) [ 1 ϵ ( x ) u m ( x ) ] + ( k 0 2 ϵ ( x ) μ m 2 ) u m ( x ) = 0 .
E z = m v m TE , E ( x ) exp ( i μ m TE y ) + v m TM , E ( x ) exp ( i μ m TM y ) ,
K z = m v m TE , K ( x ) exp ( i μ m TE y ) + v m TM , K ( x ) exp ( i μ m TM y ) .
E x = m 1 μ m TM k 0 ( k z 2 + ( μ m TM ) 2 ) 1 ϵ ( x ) v m TM , K exp ( i μ m TM y ) ,
K x = m 1 μ m TM k 0 ( k z 2 + ( μ m TE ) 2 ) v m TE , E exp ( i μ m TE y ) .
{ v m E } = { v m TE , E } { v m TM , E } ,
{ v m K } = { v m TE , K } { v m TM , K } ,
{ μ m } = { μ m TE } { μ m TM } ,
( N m E ) z = v m E ( x ) exp ( i μ m y ) ,
( N m K ) z = v m K ( x ) exp ( i μ m y ) ,
( N m E ) x = 1 μ m k 0 ( k z 2 + μ m 2 ) 1 ϵ ( x ) v m K ( x ) exp ( i μ m y ) ,
( N m K ) x = 1 μ m k 0 ( k z 2 + μ m 2 ) v m E ( x ) exp ( i μ m y ) .

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